accessible category and Ind
Proposition
Let 
be a regular cardinal and 
an infinity category.
TFAE:
- is a
-kappa accessible infinity category. 
where 
is the full infinity subcategory containing 
-compact objects, where furthermore is essentially small.- there exists a set of -compact objects such that for full subcategory
containing those objects
- there exists a small infinity category such that
see:
Proof
1) 
2)
FW: 6.62
sketch:
The inclusion
induces a factorization
(cf. universal property of Ind kappa)
one can show, that this induced map is an joyal equivalence (todo :/):
admits a right adjoint 
- happens to preserve -filtered colimits consisting of
- the counit (analogously for the unit) are equivalences on
- as generates under filtered colimits, this induces an iso on each object.
- natural equivalence between infinity functors and objectwise isomorphism
2) 3)
Let 
for some small infinity category .
Note that each object 
is a retract of some 
(where is identified with the image, cf. compact object as filtered colimit as retract)
This follows from retracts of a small subcategory as essentially small subcategory
3) 4)
special case